Download 1 - Sikkim NIC

Mathematics
1. The equation of the line parallel to 5 x  2 y  3  0 and passing through the point  1,4 is
A
C
5 x  2 y  13  0
5 x  2 y  13  0
B
D
2 x  5 y  13  0
2 x  5 y  13  0
B
D
(0, 1, 0)
(0, 0, 1)
2. The direction cosines of Y-axis are
A
C
(1, 1, 0)
(1, -1, 0)
3. The sum of the square of the first n natural numbers is
A
C
nn  12n  1
6
nn  12n  1
6
B
D
nn  12n  1
6
nn  12n  1
6
4. The area of the triangle with vertices 0,2 , 4,0 and 0,2 is
A
C
8 unit
12 unit
B
D
16 unit
none of these
5. The equation x 2  y 2  6 x  2 y  7  0 represents a
A
C
circle
ellipse
6. The value of

2
1
B
D
parabola
hyperbola
1
dx is
x
A
0
B
log 2
C
log 3
D
None of these
7. The volume of a spherical shell with radii 5 and 4 is
A
C
244
3
256
3


B
D
500
3

None of these
8. The center of the circle x 2  y 2  12 x  4 y  9  0 is
A
C
 6,2
6,2
B
D
6,2
None of these
9. The angle made by the line x  y  1 with the positive direction of the X-axis is
45
120 
A
B
150 
135 
C
D
10. Let x 2  y 2  4 x  2 y  4  0 be a circle. The point (0,0) lies
A
C
inside the circle
on the circle
B
D
outside the circle
None of these
11. The reflection of the origin with respect to the line x  y  1 is
A
B
1,1
 1,1
C
D
2,2
0,0
12. The vertex of the parabola y 2  4 x  8  0
A
C
2,0
0,2
0,2
B
D
None of these
13. When two circles touches externally to each other?
A
B
C
D
Distance between the centers is greater than the sum of their radius.
Distance between the centers is less than the sum of their radius.
Distance between the centers is equal to the sum of their radius.
None of these
14. The displacement of a moving particle is given by s  8t 2  1 , then its velocity at time t  2 sec
is
A
C
23.33 unit/sec
33 unit/sec
15. The value of lim
x

4
B
D
32 unit/sec
None of these
sin x  4 
is
x  4
A
0
B
C
1
D
1
2
None of these
B
n
D
None of these
16. The general solution of cos x  0 is
n
A
2
2n  1
C
2
2
d2y
 dy 
17. The degree of the differential equation 1     2 is
dx
 dx 
A
1
B
2
1
C
D
4
2
18. If z  1  i , then the value of Arg z  is

A
B
4
5
C
D
4
3
4
None of these.
19. The geometric mean of two real numbers a and b is
ab
A
B
2
1 1

C
D
a b
ab
2 ab
ab
20. Form a quadratic equation whose roots are 6,5 .
A
C
x 2  11x  30  0
x 2  11x  30  0
B
D
x 2  11x  30  0
x 2  11x  30  0
21. The number of combination of n objects taking r at a time is
A
C
n
Cr
nr
2
22. The eccentricity of
23. If
A
C
D
nr
B
5
4
Pr
x
y

 1 is
16 9
7
7
4
C
n
2
4
A
B
5
4
D
log 64
 x , then the value of x is
log 16
2
3
4
B
D
3
2
None of these
24. If a function f (x) is derivable at some points x  a then
A
C
It is not continuous there
The value of the function doesn’t exist
B
D
It is continuous there
none of these

2
25. The value of

0
A
C
sin x
dx is
sin x  cos x

4
1
26. If y  log(cos x) then
A
C
1
cos x
sec 2 x
B

2
D
1
B
 cos ec 2 x
D
None of these
d2y
is
dx 2
tan 1 x
0 1  x 2 dx is
1
27. The value of
A
C


2
32
B
3
16
D
None of these
2
32
1
dy

 0 , then which of the following is true:
dx
1 x2
A
B
y  sin 1 x  c
y  sin 1 x  c
C
D
None of these
x  sin 1 y  c
28. If
29. If sin x  cos x  0 , then x is
A
C
45
125 
B
D
135 
90 
B
D
cosAcosB-sinAsinB
sinAcosB-cosAsinB
30. The value of cos A  B is
A
C
sinAcosB+cosAsinB
cosAcosB+sinAsinB
2 x  3, x  0
31. If f x   
, then the function is
2 x  3, x  0
A
continuous everywhere
B
C
continuous except at x  0 D
continuous no-where
continuous only at x  0
32. The distance of the point A2,0 from the line x  y  1 is
1
A
2
B
2
C
D
None of these
2


33. The value of cos  315 is
1
A
2
3
C
2
B
D

1
2
3

2
34. The value of tan 1 tan 2  tan 3  tan 88  tan 89  is
A
C
0
1
B
D
-1
None of these


35. The principle value of the argument of the complex number 3  i , i   1 is

5
A
B
6
6
11
C
D
None of these
6
36. If  and  be the roots of the equation x 2  5 x  1  0 , then the value of  2   2 is
A
C
25
23
B
D
27
None of these
37. The square root of i is
A
C
1  i 
 12 1  i 

1
2
1  i 
 12 1  i 

B
D
1
2



38. If w is the imaginary cube root of unity, then the value of 1  w7  w8 1  w 4  w5 is
A
C
0
 4
39. The value of the determinant A 
A
C
1
0
4
None of these
B
D
cos x
sin x
 sin x cos x
B
D
is
-1
sin xcos x
40. In how many ways can the result of three successive football matches between Brazil and
Argentina be decided?
A
C
9
27
B
D
3
81
a
 f xdx , where f x is an even function
41. The value of
A
a
a
2
is
B
0
D
does not exist
a
C
2 f x dx
0
 2 1
1 3 0
 and B  
 , then which of the following statement is FALSE:
42. If A  
5
3
2
5
7




A
AB exists
B
BA exists
1
C
D
A exists
B 1 does not exist
43. The value of log 4 log 25 9  log 9 25 is
A
C
0
2
B
D
1
None of these
44. The value of lim 1  x  x is
1
x 0
A
C
e
1
B
D
1
e
0
45. In how many ways can five persons be seated around a circular table?
A
25
B
24
C
20
D
120
46. The expression for log e (1  x) is
A
B
C
D
x x 2 x3
   ..................
1! 2! 3!
x x 2 x3
1     ..................
1 2 3
x 2 x3 x 4
x     ...............
2 3 4
None of these
1
3x, 0  x  2
47. Let f x   
, if f x  is continuous at x  2 then  is
 , 2  x  3
A
3
B
0
C
-6
D
6
48. the 15th term of the series 1  4  7  10   is
A
C
46
44
B
D
45
43
x 3  27
is
x 3 x  3
does not exist
0
B
D
27
9
49. The value of lim
A
C
50. If z be any complex number, then z  z is
A
C
Purely real
Zero
B
D
Purely imaginary
None of these
51. How many equations are required to represent a curve in space?
A
C
1
3
B
D
2
None of these
52. The ratio of the perimeter to its diameter of a circle is
A
C

3
B
D
2
None of these
53. A coin is tossed four times. In how many different ways can we obtain one head and three tails?
A
C
1
3
B
D
2
None of these
54. In how many ways can a committee of eight be chosen from ten individuals?
10
10
P8
C8
A
B
C
D
None of these
80
55. The distance, s traversed by a particle in a straight line from the origin, at any time t is given by
s  2t 2  3t . The velocity of the particle at 4 second is
A
C
19
20
B
D
44
None of these

56. Find the magnitude of the vector a  2iˆ  3 ˆj  kˆ .
A
C
6
14
B
D
14
None of these
57. How many seven digits telephone numbers are possible, if only odd digits may be used?
7
7
P5
C5
A
B
57
C
D
75
58. The total number of subsets of a set having n elements is
n2
n
P2
A
C
B
D
2n
n
C2
59. The sum of the square of the direction cosines of a Straight line is
A
C
0
2
B
D
1
None of these
log e tan x
log e sec x  tan x 
B
log e sec x  tan x 
tan x
60.  sec xdx 
A
C
 esin x
61.  
 1 x2

1
A
C
D

dx is


1
esin x
1
e cos ec x
B
D
sin 1 xesin
1
x
None of these
62. If a set A has 3 elements ,the total number of functions from A to itself is
A
C
9
1
B
D
6
none of these
63. The family of straight lines passing through the origin is represented by the differential equation
ydx  xdy  0
xdx  ydy  0
A
C
xdy  ydx  0
ydy  xdx  0
B
D
64. A line makes an angles  ,  ,  with the co-ordinate axes ,then cos 2   cos 2   cos 2  
A
0
B
1
C
-1
D
none of these
65. A matrix A is invertible (the inverse of a matrix exist) if its determinant value is
A
C
0
non zero
B
D
1
none of these
66. The sum of the series 2  4  6   up to 20 terms is
A
C
440
420
B
D
320
340
67. Extreme value of f ( x)  x 2  x is at x 
A
0
B
1
2
C
1
D

1

2
1
2
68. Maximum possible value of sin  is
A
0
B
C
-1
D
69. The tangent to a curve touches it at
A
C
only one point
infinite number of points
B
D
two points
none of these
70. If lim f x  exists then which one of these is true
x a
A
B
C
D
f (x)
f (x)
f (x)
f (x)
dy
 0 at
dx
tangent at x  2
tangent at x  2
tangent at x  2
none of these
71. If for y  f (x) ,
A
B
C
D
is always continuous at x  a
is always derivable at x  a
is always discontinuous at x  a
is continuous at x  a if lim f x   f a 
xa
x  2 then which one of these is true
to y  f (x) is parallel to X axis
to y  f (x) is parallel to Y axis
to y  f (x) does not exist
72. A person can arrange 5 books on a shelf in
A
C
5 ways
24 ways
B
D
120 ways
25 ways
73. The number of three digit numbers formed from the digits 0,1,2,3 is
A
C
64
24
B
D
48
12
1 4 8 
74. The determinant value of the matrix A  0 3 6  is
0 2 4
A
C
1
0
B
D
-1
12
 2 1
5 0
 and B  
 , then AB  I 
75. If A  
 0 3
 4 2
15 12 


A
B
2 7
C
3
76. If A  
5
A
C
7 2


12 15 
15 2 


12 7 
12 7 


15 2 
D
5
 , then which of the following statement is FALSE
2 
A is a symmetric matrix
B
transpose of A is A itself
A is a anty-symmetric matrix
D
None of these
77. If f x   x 2 and g x   e x then which of the following statement is TRUE
A
B
C
D
f g x  g  f x
f g x  g  f x
f g x and g  f x may or may not be same
None of these
78. If f x  sin x and g x  log x then the value of g  f x at x 
A
C
0
-1

is
2
B
D
1
None of these
B
D
log b a
79. If a x  b then x 
A
C
log a b
log e ab
None of these
3 
1 0
 2x  1
3
 , A  
 and B  
80. If 2 A  3B  I , where I  
2 y  1
0 0
 6
4
A
B
x  0 and y  2
x2
C
D
x  2 and y  2
x0
81. If A  x : 0  x  5 and B  x : 2  x  7 then A  B 
A
C
x : 2  x  5
x : 2  x  5
2
 then
2 
and y  0
and y  0
x : 2  x  5
B
D
None of these
B
D
8
4
82. If a  2i  3 j  k and b  i  2 j  4k then a.b 
A
C
83.
10
6
d
(Sec-1x) is
dx
1
A
B
x 1
1
2
C
D
x2 1

84. The value of
1
 n!
1
x x2 1
x
x2 1
is
n 0
A
C
85.
1
e
1
B
e
D
None of these
d x
( x ) is
dx
A
e x log x
B
C
x x (1  log x)
D
1
log x
x
x x (1  log x)
86. Divide the number 15 into three such parts that the may form an A.P., and that the product of
the first two parts may be 80. The parts are
A
C
2, 5, 8
3, 5, 7
B
D
1, 5, 9
None of these
B
D
x  0, y  0
None of these
87. The equation of Z-axis is
A
C
z 0
x yz
88. Which term of the G.P., 2, 6, 18, 54… is 1458?
A
C
9th term
7th term
B
D
8th term
6th term
B

D
None of these
B
D
1
does not exist
89. The value of sin 1 x  cos 1 x is
A
C
0

2
dy
at x  0 is
dx
90. If y  e x cos x then
A
C
-1
0


91. The differential equation e x  1 dy   y  1e x dx has the solution
A
C

1  y e x  2  c
92. The domain of f ( x) 
A
C

y 1  c ex 1
B
D
1  y e x  1  c
1  y e x  1  c
B
D
{x | - 4  x  4}
{x | 0  x  4}
1
is
x4
{x | x  4}
{x | x  4}
93. In a ABC , if a  18, b  24, c  30 then its area is
A
C
96 sq. units
216 sq. units
B
D
612 sq. units
None of these
1 2  3 8 
  
 ?
94. What must be the matrix X if 2X  
3 4  7 2
1 3 


A
B
 2 1
C
2 6 


 4  2
 1  3


 2  1
 2  6


 4  2
D
95. If A and B are two matrices such that A+B and AB are both define, then A and B are
A
C
both null matrices
both square matrices of same order
B
D
both identity matrices
None of these
96. The value of lim
x 3
A
C
x3
is
x3
0
-1
B
D
1
does not exist
97. The function f x   x 3 is
A
increasing only in (0, 1)
B
C
every where increasing
D
2
2
x
y
98. The area of the ellipse 2  2  1 in the first quadrant is
a
b

A
B
4ab
ab
C
D
4
ab
2
None of these
3
2
 dy   d y 
99. The order of the differential equation     2   0 is
 dx   dx 
A
2
B
C
6
D
2
decreasing in(0, 1)
every where decreasing
3
1
100. The value of k for which 3x 2  8xy  ky 2  0 represents two perpendicular lines is
A
3
B
-3
3
2
C
D
2
3